Let $A[[h]]$ be the formal power series algebra over $\mathbb{C}[[h]]$, here $\mathbb{C}$ is the complex number field. Is the canonical map $A[[h]] \otimes_{\mathbb{C}[[h]]} A[[h]] \to (A\otimes_\mathbb{C} A)[[h]]$ injective? Thanks!
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What canonical map? – egreg Apr 12 '15 at 14:28
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Hi, egreg. The canonical map is sending \sum_{i=0}^{\infty}a_i h^i\otimes \sum_{j=0}^{\infty}b_j h^j to the Cauchy product \sum_{n=0}^{\infty}\sum_{n=i+j}a_i b_j h^n. – wsq Apr 16 '15 at 12:06
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The codomain in your comment doesn't match the question's. – egreg Apr 16 '15 at 12:50
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O.K. Let me write the map precisely. It sends $ \sum\limits_{i=0}^{\infty}a_i h^i\otimes \sum\limits_{j=0}^{\infty}b_j h^j$ to $\sum\limits_{n=0}^{\infty}(\sum\limits_{i=0}^{n}a_i \otimes b_{n-i}) h^n.$ – wsq Apr 22 '15 at 12:46
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@egreq Thanks for your reply. – wsq Apr 22 '15 at 12:50
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Is that map well defined? – egreg Apr 22 '15 at 12:59
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Yes,it is. One can check this by using the universal property of tensor product. – wsq Apr 24 '15 at 11:31